Emerging　applications　in　machine　learning　have　imposed　the　problem　of　monotone　non-submodular　maximization　subject　to　a　cardinality　constraint.　Meanwhile,　parallelism　is　prevalent　for　large-scale　optimization　problems　in　bigdata　scenario　while　adaptive　complexity　is　an　important　measurement　of　parallelism　since　it　quantifies　the　number　of　sequential　rounds　by　which　the　multiple　independent　functions　can　be　evaluated　in　parallel.　For　a　monotone　non-submodular　function　and　a　cardinality　constraint,　this　paper　devises　an　adaptive　algorithm　for　maximizing　the　function　value　with　the　cardinality　constraint　through　employing　the　generic　submodularity　ratio　gamma　to　connect　the　monotone　set　function　with　submodularity.　The　algorithm　achieves　an　approximation　ratio　of　1　-　e-(gamma　2)　-　epsilon　and　consumes　O(log(n/eta)/epsilon(2))　adaptive　rounds　and　O(n　log　log　(k)/epsilon(3))　oracle　queries　in　expectation.　Furthermore,　when　gamma=1,　the　algorithm　achieves　an　approximation　guarantee　1　-　1/e　-　epsilon,　achieving　the　same　ratio　as　the　state-of-art　result　for　the　submodular　version　of　the　problem.